Metal disk problem!

NasuSama

1. The problem statement, all variables and given/known data

A uniform metal disk (M = 8.21 kg, R = 1.88 m) is free to oscillate as a physical pendulum about an axis through the edge. Find T, the period for small oscillations.

2. Relevant equations

[itex]I = mr^{2}/4[/itex]
[itex]T = 2\pi √(I/mgd)[/itex]

3. The attempt at a solution

I combined the formula together to get:

[itex]T = 2\pi √((mr^{2}/4)/(mgr))[/itex]
[itex]T = 2\pi √(r/(4g))[/itex]

But the answer is incorrect

Doc Al

How did you arrive at this result?

NasuSama

I am thinking that I need to use the moment of inertia of the disk.

Doc Al

Of course you do, but that's not the correct formula.

NasuSama

Then, it's something like I = mr²/2, rotating to its center. However, the disk oscillates through its edge.

I am not sure which path to go for...

Doc Al

Right.
Use the parallel axis theorem. (Look it up!)

NasuSama

Hm.. By the Parallel Axis Theorem, I would assume that:

[itex]I = I_{center} + md^{2}[/itex]
[itex]I = mr^{2}/2 + mr^{2}[/itex] [Since the disk rotates about an axis through the edge, we must add the inertia by mr². r is the distance between the center and the edge of the disk.]
[itex]I = 3mr^{2}/2[/itex]

Is that how I approach this? Let me know where I go wrong. Otherwise, I can just plug and chug this expression:

[itex]T = 2\pi √((3mr^{2}/2)/(mgr))[/itex]
[itex]T = 2\pi √(3r/(g))[/itex]

NasuSama

Nvm. My answer is right. Thanks for your help by the way!

Doc Al

Good!